VATICAN CITY — Bishops from around the world began discussing proposals Thursday for running the church, including ways to address the priest shortage and how to resolve the problem of giving Communion to Catholics who divorce and remarry.

While it seems the Synod of Bishops won’t recommend relaxing the celibacy rule for priests, consensus seemed to be growing to recommend that church tribunals speed up annulment processes so Catholics can remarry in church and thus be allowed to receive Communion.

Church teaching holds that Catholics who divorce and remarry without getting an annulment cannot receive the sacrament because their condition “objectively contrasts with God’s law.” An annulment is a church ruling that the original marriage was invalid.

The priest shortage and the situation of Catholics who are denied Communion have been key themes of the Oct. 2-23 meeting, which is designed to let bishops from around the world vote on proposals that are then sent to Pope Benedict XVI to consider in a future document.

Thursday was the first full day in which the bishops met in small groups to begin crafting proposals, after having heard speeches from all the 250 bishops and other experts gathered.

Cardinal Angelo Scola, the key moderator of the meeting, has emphasized the need for church tribunals to be more efficient, saying in a summary report to the closed-door meeting Wednesday that several bishops had suggested the procedures be simplified or that tribunals be created where they don’t exist.

Pope John Paul II, however, had complained that annulments were too easily obtained and expressed worry that tribunals face the risk of corruption.

Cardinals dismissed the apparent contradiction Thursday.

“It’s not a deception,” said Cardinal Francis Arinze, a synod official. “What John Paul II said was that these tribunals must work according to truth, and not according to Oh, these people are suffering so and let us declare it invalid.'”

“What is being asked for is that the bureaucracy of the tribunals not be heavy and that the cases be resolved in a more just time to help these people,” Cardinal Juan Sandoval Iniguez of Mexico said.

Archbishop John Foley, who heads the Vatican’s social communications office, said it was too soon to say whether a specific proposal on making the tribunals more efficient would emerge from the synod. But he said the seeming consensus that appeared to be emerging was a correct “reading” of the trend.

Several synod participants who appeared at a news conference Thursday concurred that the priest shortage was among the major synod issues. But they dismissed reporters’ questions about whether married priests were an option, saying a celibate priesthood was not the reason for the shortage.

“The shortage of priests is a symptom of the problem,” said Cardinal Telesphore Placidus Toppo of India. “The real problem is the crisis of faith.”

In his summary, Scola listed 17 questions for the bishops to consider as they meet in small discussion groups to draft proposals for the pope, including how they should respond to the “urgent duty” to offer the Eucharist to all the faithful, “even in mission countries and where there is a scarcity of priests.”

Scola noted that several bishops had raised the issue, including references to so-called “viri probati” or married men of proven virtue who could be ordained.

Others spoke of the need to better distribute the priests that exist, while several bishops of the Eastern rite, which allows married priests, spoke about their experiences.

“Various Eastern fathers referred to the practice of married priests in their churches, offering each one of us elements for a further careful evaluation of the choice of the Latin church to connect celibacy to ordained priesthood,” Scola said, according to the speech released Thursday by the Vatican.

“To this proposal, some fathers … have affirmed that the hypothesis of the viri probati’ wasn’t a path worth traveling.”

Monsignor Sofron Stefan Mudry, of the Eastern rite Catholic church in Ukraine, echoed this sentiment at the news conference, saying a married priesthood brings “unending problems” to the church, including housing issues and the costs involved in supporting a priest with a family.

He said of his 400 priests, 360 are married, even though in the past, the Greek Catholic Church in Ukraine had aimed to keep an equal number of married and celibate priests.